This work provides a Java program which constructs free polyominoes of size n sorted by width
and height of the convex hull (i.e., its rectangular bounding box). The results correct counts
for 15-ominoes published in the 1967 proceedings of the SIAM Fall Meeting, and
extend them to 16-ominoes and partially to even larger polyominoes.
Category:Combinatorics and Graph Theory

A New Algebraic Approach to the Graph Isomorphism and Clique Problems

Authors:Roman GalayComments: 7 Pages. The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually this fundamental principle can be efficiently applicable in Computational Mathematics and Complexity Theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) on a class of instances is often too difficult to determine, although we may also assume the existence of a variety of algorithms for NP-complete problems that are both correct and polynomial-time on all the instances from a given class (where the given problem remains NP-complete), but whose correctness or/and polynomial-time complexity on the class is impossible to prove as an example for Gödel's theorems. However, supposedly such algorithms should possess a certain complicatedness of processing the input data and treat it in a certain algebraically “entangled” manner. The same algorithmic analysis in fact concerns all the other significant problems and subclasses of NP, such as the graph isomorphism problem and its associated complexity class GI.
The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.
Category:Combinatorics and Graph Theory

Introducing: Second-Order Permutation

In this study we answer questions that have to do with finding out the total number of ways of arranging a finite set of symbols or objects directly under a single line constraint set of finite symbols such that common symbols between the two sets do not repeat on the vertical positions. We go further to study the outcomes when both sets consist of the same symbols and when they consist of different symbols. We identify this form of permutation as 'second-order permutation' and show that it has a corresponding unique factorial which plays a prominent role in most of the results obtained. This has been discovered by examining many practical problems which give rise to the emergence of second-order permutation. Upon examination of these problems, it becomes clear that a directly higher order of permutation exist. Hence this study aims at equipping mathematics scholars, educators and researchers with the necessary background knowledge and framework for incorporating second-order permutation into the field of combinatorial mathematics.
Category:Combinatorics and Graph Theory

Even FibBinary Numbers and the Golden Ratio

Previously, a determination of the relationship between the Natural numbers (N) and the n'th odd fibbinary number has been made using a relationship with the Golden ratio \phi=(Sqrt[5]+1}/2 and \tau=1/\phi. Specifically, if the n'th odd fibbinary equates to the j'th N, then j=Floor[n(\phi+1) - 1]. This note documents the completion of the relationship for the even fibbinary numbers, such that if the n'th even fibbinary equates to the j'th N, then j=Floor[n(\tau+1) + \tau].
Category:Combinatorics and Graph Theory

The Clique Problem - A Polynomial Time and Non-Heuristic Solution

The Clique Problem
This paper provides a Polynomial Time and Non-Heuristic Solution to the Clique problem.
Methods are given to find:
1.Maximum clique (a clique with the largest possible number of vertices),
2.Listing all maximal cliques (cliques that cannot be enlarged), and
3.Solving the decision problem of testing whether a graph contains a clique larger than a given size.
4.Finding cliques of a selected size, particularly largest cliques.
Category:Combinatorics and Graph Theory

The 2n-2 Lines Proof of the n x n Points Puzzle

This paper offers the first proof that the minimal solution of the n x n dots puzzle, for any n≥3, counts 2n-2 lines. Furthermore, a general criterion to solve any n x n grid is given.
Category:Combinatorics and Graph Theory

The Permanent and Diagonal Products on the Set of Nonnegative Matrices with Bounded Rank

We formulate conjectures regarding the maximum value and maximizing matrices
of the permanent and of diagonal products on the set of stochastic matrices with
bounded rank. We formulate equivalent conjectures on upper bounds for these func-
tions for nonnegative matrices based on their rank, row sums and column sums.
In particular we conjecture that the permanent of a singular nonnegative matrix is
bounded by 1/2 times the minimum of the product of its row sums and the product of
its column sums, and that the product of the elements of any diagonal of a singular
nonnegative matrix is bounded by 1/4 times the minimum of the product of its row
sums and the product of its column sums.
Category:Combinatorics and Graph Theory

The n X n X n Dots Problem: An Improved “Outside the Box” Upper Bound

In this paper we describe two new patterns, in order to improve the upper bound for the Ripà’s n X n X n points problem, saving a few lines for many values of n. The new upper bound, for any n≥6, becomes h_u(n)=int((3/2*n^2)+int(n/4)-int((n-1)/4)+int((n+1)/4)-int((n+2)/4)+n-2, where int(x)≔floor(x).
Category:Combinatorics and Graph Theory

Interval Complex Neutrosophic Graph of Type 1

The neutrosophic set theory, proposed by smarandache, can be used as a general mathematical tool for dealing with indeterminate and inconsistent information. By applying the concept of neutrosophic sets on graph theory, several studies of neutrosophic models have been presented in the literature. In this paper, the concept of complex neutrosophic graph of type 1 is extended to interval complex neutrosophic graph of type 1(ICNG1). We have proposed a representation of ICNG1 by adjacency matrix and studied some
properties related to this new structure. The concept of ICNG1 generalized the concept of generalized fuzzy graphs of type 1 (GFG1), generalized single valued neutrosophic graphs of type 1 (GSVNG1) generalized interval valued neutrosophic graphs of type 1 (GIVNG1) and complex neutrosophic graph type 1(CNG1).
Category:Combinatorics and Graph Theory

Independence of Clique Problems

A polynomial algorithm is “faster” than an exponential algorithm. As n grows an (exponential) always grows faster than nk (polynomial), i.e. for any values of a and k, after n> certain integer n0, it is true that an > nk. Even 2^n grows faster than n1000 at some large value of n. The former functions are exponential and the later functions are polynomial. It seems that for some problems we just may not have any polynomial algorithm at all (as in the information theoretic bound)! The theory of NPcompleteness is about this issue, and in general the computational complexity theory addresses it.
Category:Combinatorics and Graph Theory

The study of perebor dates back to the Soviet-era mathematics, especially in the 1980s [1]. Post-Soviet mathematicians have been working on many problems in combinatorial optimization. One of them is Maximum Edge Biclique Problem (MBP). In [2], the author proves that MBP is NP-complete. In this note, we give a polynomial time algorithm for MBP by using linear programming (LP). Thus, some flaw needs to be found in Peeter's work. We leave this to the community.
Category:Combinatorics and Graph Theory

Exact Weight Perfect Matching of Bipartite Graph Problem Simplified

The study of perebor dates back to the Soviet-era mathematics, especially in the 1980s [1]. Post-Soviet mathematicians has been working on many problems in combinatorial optimization. One of them is Exact Weight Perfect Matching of Bipartite Graph (EWPM).This particular problem has been thoroughly considered by [2], [3], [4]. In this note, we give a simpler proof about the solvability of EWPM.
Category:Combinatorics and Graph Theory

Proof of the Goldbach Conjecture

This proves that any even number larger than 2 can be written as the sum of two prime Numbers, also known as the "goldbach conjecture" or "goldbach conjecture about the even" is in the test for any greater than or equal to 6 even conform to guess the number of prime Numbers, accidentally discovered the prime Numbers of "additionality" and further expansion of verification.This article does not focus on the functional expressions of prime Numbers themselves, but takes a different approach to prove that all even Numbers can be composed of two prime Numbers
Category:Combinatorics and Graph Theory

CSP Solver and Capacitated Vehile Routing Problem

In this paper, we present several models for Capacitated Vehicle Routing Problem (CVRP) using Choco solver. A concise introduction to the constraint programming methods is included. Then, we construct two models for CVRP. Experimental results for each model are given in details.
Category:Combinatorics and Graph Theory

Envp, Another Prime Number Based Strategy to Encode Graphs

Abstract: In this paper we show a method to encode graphs with a numerical value that follows unique labeling of each vertex or node and unique labeling of each edge of a graph with unique prime numbers. Each edge is defined as the connectivity between two vertices, therefore two vertices or nodes connected by an edge may be represented by the “ edge-nodes value ” derived by raising the prime number representing the edge to the product of the primes representing the two nodes that are connected by that edge. Multiplying all the “edge-nodes values” of a single graph will represent a unique number albeit very large in majority of cases. Given this unique number called the “Edge-nodes values product”, it is possible to derive the structure of the given graph. This encoding may allow new approaches to graph isomorphism, cryptography, quantum computing, data security, artificial intelligence, etc.
Category:Combinatorics and Graph Theory

Labeled Trees with Fixed Node Label Sum

The non-cyclic graphs known as trees may be labeled by assigning
positive integer numbers (weights) to their vertices or to their edges.
We count the trees
up to 10 vertices that have prescribed sums of weights, or, from
the number-theoretic point of view, we count the compositions
of positive integers that are constrained by
the symmetries of trees.
Category:Combinatorics and Graph Theory

The aim of this article is to introduce a matrix algorithm for finding minimum spanning tree (MST) in the environment of undirected bipolar neutrosophic connected graphs (UBNCG). Some weights are assigned to each edge in the form of bipolar neutrosophic number (BNN). The new algorithm is described by a flow chart and a numerical example by considering some hypothetical graph. By a comparison, the advantage of proposed matrix algorithm over some existing algorithms are also discussed.
Category:Combinatorics and Graph Theory

Rozpatrywane w tej pracy rodzaje grafów można utworzyć na podstawie równań stechiometrycznych. Główną zaletą metod grafowych jest możliwość uzyskania informacji o stabilności stanów stacjonarnych bez wypisywania w jawnej postaci jakichkolwiek równań.
###
The types of graphs considered in this work can be created on the basis of stoichiometric equations. The main advantage of the graph theory methods is the possibility of obtaining information on the stability of steady states without writing out any equations in an explicit form.
Category:Combinatorics and Graph Theory

On the Division of Planar Graphs in Consecutive Prime Parts

In this paper it is discussed the following problem: "A mathematician
wants to divide is garden into consecutive prime parts (first in two parts,
after in three parts, and so on), only making straight paths, in a simple
way (without retracing his own steps), and without going out of his plot
of land. In how many parts can the mathematician divide his garden?"
Category:Combinatorics and Graph Theory

Cause/effect Correlations Through the Borsuk-Ulam Theorem and Kneser Graphs

The assessment of hidden causal relationships, e.g., adverse drug reactions in pharmacovigilance, is currently based on rather qualitative parameters. In order to find more quantifiable parameters able to establish the validity of the alleged correlations between drug intake and onset of symptoms, we introduce the Borsuk-Ulam Theorem (BUT), which states that a single point on a circumference projects to two points on a sphere. The BUT stands for a general principle that describes issues from neuroscience, theoretical physics, nanomaterials, computational topology, chaotic systems, group theory, cosmology. Here we introduce a novel BUT variant, termed operational-BUT, that evaluates causal relationships. Further, we demonstrate that the BUT is correlated with graph theory and in particular with the so-called Kneser graphs: this means that the combinatory features of observables, such as the bodily responses to drug intake, can be described in terms of dynamical mappings and paths taking place on well-established abstract structures. Therefore, physical and biological dynamical systems (including alleged causes and their unknown effects) make predictable moves into peculiar phase spaces, giving rise to
constrained trajectories that can be quantified.
Category:Combinatorics and Graph Theory

The Exact Solution of Gauss’s Problem on the Number of Integer "Points" in a Circular and Spherical "Layers"

In the article, the Gauss’s problem on the number of integer points for a circle and a ball in the framework of an integer lattice is reformulated in an equivalent way and reduces to solving two combinatorial tasks for a circular and spherical "layer" in the framework of Quantum Discrete Space. These tasks are solved using trigonometric functions defined on a set of integers whose range of values is also integers, and other new mathematical tools. It comes not about evaluative solutions, but about exact solutions, which, if necessary, can be transferred to a circle and a ball. In doing so not only specific formulas for determine the exact number of solutions are presented, but also the formulas for enumerating the corresponding pairs and triples of integers. The importance of obtained solutions lies in the fact that they determine the analytical likenesses of not only the circumference and the sphere in the Quantum Discrete Space, but also point to the possibility of constructing of the likenesses of ellipse, cone, hyperboloid and other figures.
Category:Combinatorics and Graph Theory

A Note on Vertex Transitivity in Isomorphic Graphs

Authors:Murugesan, AnithaComments: 9 Pages. The last theorem is the highlight of the paper

In the graph theory, two graphs are said to be isomorphic if there is a one-one, onto mapping defined between their set of vertices so as to preserving the adjacency between vertices. An isomorphism defined on a vertex set of a graph to itself is called automorphism of the given graph. Two vertices in a graph are said to be similar if there is an automorphism defined on its vertex set mapping one vertex to the other. In this paper, it has been discussed that every such automorphism defines an equivalence relation on the set of vertices and the number of equivalence classes is same as the number of rotations that the automorphism makes on the vertex set. The set of all automorphisms of a graph is a permutation group under the composition of permutations. This group is called automorphism group of the graph. A graph is said to be vertex transitive if its automorphism group acts transitively on its vertex set. The path degree sequence of a vertex in a graph is the list of lengths of paths having this particular vertex as initial vertex. The ordered set of all such sequences is called path degree sequence of the graph. It is conjectured that two graphs are isomorphic iff they have same path degree sequence. In this paper, it has been discussed that this conjecture holds good when both the graphs are vertex transitive. The notion of functional graph has been introduced in this paper. The functional graph of any two isomorphic graphs is a graph in which the vertex set is the union of vertex sets of isomorphic graphs and two vertices are connected by an edge iff they are connected in any one of the graph when they belong to the same graph or one vertex is the image of the other under the given isomorphism when they are in different set of vertices. It has been proved that the functional graphs obtained from two isomorphic complete bipartite graphs are vertex transitive.
Keywords : graph automorphism ; functional graph ; vertex transitive graph ; path degree sequence.
Category:Combinatorics and Graph Theory

In this paper, motivated by the notion of
generalized single valued neutrosophic graphs of first type, we
defined a new neutrosophic graphs named generalized interval
valued neutrosophic graphs of first type (GIVNG1) and
presented a matrix representation for it and studied few
properties of this new concept. The concept of GIVNG1 is an
extension of generalized fuzzy graphs (GFG1) and generalized
single valued neutrosophic of first type (GSVNG1).
Category:Combinatorics and Graph Theory

In this article, we present an algorithm method
for finding the shortest path length between a paired nodes on a
network where the edge weights are characterized by single
valued triangular neutrosophic numbers. The proposed
algorithm gives the shortest path length from source node to
destination node based on a ranking method. Finally a numerical
example is presented to illustrate the efficiency of the proposed
approach
Category:Combinatorics and Graph Theory

In this paper, we introduced a new neutrosophic graphs called complex neutrosophic graphs of type1 (CNG1) and presented a matrix representation for it and studied some properties of this new concept. The concept of CNG1 is an extension of generalized fuzzy graphs of type 1 (GFG1) and generalized single valued neutrosophic graphs of type 1 (GSVNG1).
Category:Combinatorics and Graph Theory

Statistics on Small Graphs

We create the unlabeled or vertex-labeled graphs with up to 10
edges and up to 10 vertices and classify them by a set of standard properties:
directed or not, vertex-labeled or not, connectivity, presence of isolated vertices, presence of multiedges and presence of loops. We present tables of how
many graphs exist in these categories.
Category:Combinatorics and Graph Theory

Traveling Salesman Problem Solved with Zero Error

The traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The general approach to solving the different types of NP problems is the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. In the salesman problem, the first step is to arrange the data in the problem in increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The approach in this paper is different from the author's previous approach (viXra:1505.0167) in which the needed distances not among the least ten distances were added to the least ten distances before route construction began. In this paper, one starts with only the least ten distances and only if a needed distance is not among the set of the least ten distances, would one consider distances greater than those in the set of the ten least distances.
The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in workforce project management and hiring, as well as in a country's workforce needs and immigration quota determination. Since an approach that solves one of these problems can also solve other NP problems, and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied
Category:Combinatorics and Graph Theory

The nxnxn Dots Problem Optimal Solution

Authors:Marco RipàComments: This is a revised version of the paper published in 2016 on Notes on Number Theory and Discrete Mathematics (ISSN 1310-5132), Volume 22, Number 2 (Pages 36—43).

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach, that significantly reduces the number of straight lines connected at their end-points necessary to join all the n^3 dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), for any n > 1.
Category:Combinatorics and Graph Theory

Decision-Making Method based on Neutrosophic Soft Expert Graphs

In this paper, we first define the concept of neutrosophic soft expert graph. We have established a link between graphs and neutrosophic soft expert sets. Basic operations of neutrosophic soft expert graphs such as union, intersection and complement are defined here. The concept of neutrosophic soft expert soft graph is also discussed in this paper. The new concept is called neutrosophic soft expert graph-based multi-criteria decision making method (NSEGMCDM for short). Finally, an illustrative example is given and a comparison analysis is conducted between the proposed approach and other existing methods, to verify the feasibility and effectiveness of the developed approach.
Category:Combinatorics and Graph Theory

New Lower Bounds for Van Der Waerden Numbers Using Genetic Algorithm

Genetic algorithm is a good tool for finding the global minimum in many discrete problems. In particular, it has proven itself in problems where there is no any apriori information about the possibilities of narrowing the search, or the specifics of the problem do not involve such. This work describes the procedure of using a genetic algorithm as applied to the search of van der Waerden numbers. Some new lower bounds of van der Waerden numbers were found using this procedure.
Category:Combinatorics and Graph Theory

A Neutrosophic Graph Similarity Measures

This paper is devoted for presenting new neutrosophic similarity measures between
neutrosophic graphs. We proposetwo ways to determine the neutrosophic distance between
neutrosophic vertex graphs. The two neutrosophic distances are based on the Haussdorff distance,and a robust modified variant of the Haussdorff distance, moreover we show that they both satisfy the metric distance measure axioms. Furthermore, a similarity measure between neutrosophic edge graphs that is based on a probabilistic variant of Haussdorff distance is introduced. The aim is to use those measures for the purpose of matching neutrosophic graphs whose structure can be described in the neutrosophic domain.
Category:Combinatorics and Graph Theory

Foundation for Neutrosophic Mathematical Morphology

The aim of this paper is to introduce a new approach to Mathematical Morphology based on
neutrosophic set theory. Basic definitions for neutrosophic morphological operations are extracted and a study of its algebraic properties is presented. In our work we demonstrate that neutrosophic morphological operations inherit properties and restrictions of Fuzzy Mathematical Morphology
Category:Combinatorics and Graph Theory

Spectra of New Join of Two Graphs

Let G1 and G2 be two graph with vertex sets V (G1); V (G2) and
edge sets E(G1);E(G2) respectively. The subdivision graph S(G) of a graph
G is the graph obtained by inserting a new vertex into every edges of G. The
SGvertexjoin of G1 and G2 is denoted by G1}G2 and is the graph obtained
from S(G1) [ G1 and G2 by joining every vertex of V (G1) to every vertex
of V (G2). In this paper we determine the adjacency spectra ( respectively
Laplacian spectra and signless Laplacian spectra) of G1}G2 for a regular graph
G1 and an arbitrary graph G2
Category:Combinatorics and Graph Theory

Spectra of a New Join in Duplication Graph

The Duplication graph DG of a graph G, is obtained by inserting
new vertices corresponding to each vertex of G and making the vertex adja-
cent to the neighbourhood of the corresponding vertex of G and deleting the
edges of G. Let G1 and G2 be two graph with vertex sets V (G1) and V (G2)
respectively. The DG - vertex join of G1 and G2 is denoted by G1 t G2 and
it is the graph obtained from DG1 and G2 by joining every vertex of V (G1)
to every vertex of V (G2). The DG - add vertex join of G1 and G2 is denoted
by G1 ./ G2 and is the graph obtained from DG1 and G2 by joining every
additional vertex of DG1 to every vertex of V (G2). In this paper we determine
the A - spectra and L - spectra of the two new joins of graphs for a regular
graph G1 and an arbitrary graph G2 . As an application we give the number
of spanning tree, the Kirchhoff index and Laplace energy like invariant of the
new join. Also we obtain some infinite family of new class of integral graphs
Category:Combinatorics and Graph Theory

Spectrum of (K; r) Regular Hypergraph

We present a spectral theory of uniform, regular and linear hyper-
graph. The main result are the nature of the eigen values of (k; r) - regular
linear hypergraph and the relation between its dual and line graph. We also
discuss some properties of Laplacian spectrum of a (k; r) - regular hypergraphs.
Category:Combinatorics and Graph Theory

The Premature State of “Topology”and “Graph Theory” Nourished by “Seven Bridges of Königsberg Problem”

In this paper, we'll be discussing about the "Seven Bridges of Königsberg Problem". This paper will help you to understand, "How Euler solved this problem?". And, It will give some intuition of "Topology" and "Graph Theory".
Category:Combinatorics and Graph Theory

Tiling Hexagons with Smaller Hexagons and Unit Triangles

This is a numerical study of the combinatorial problem of packing hexagons of some equal size
into a larger hexagon. The problem is well defined if all hexagon edges have integer length
and if their centers and vertices share the common lattice points of a triangular grid with unit distances.
Category:Combinatorics and Graph Theory

An Algorithm for Solving the Graph Isomorphism Problem

This article presents an algorithm for solving the graph isomorphism problem. Under certain circumstances the algorithm is definitely polynomial time, and it could possibly always be polynomial time, but that hasn't been verified. The algorithm also hasn't been tested on graphs with more than three nodes, nor has it been reviewed by anyone so far.
Category:Combinatorics and Graph Theory

Intuition-Based ai for Solutions of NP-Complete Problems.

The challenge of this paper is to relate artificial intuition-based intelligence, represented by self-supervised systems, to solutions of NP-complete problems. By self-supervised systems we understand systems that are capable to move from disorder to order without external effort, i.e. in violation of the second law of thermodynamics. It has been demonstrated, [1], that such systems exist in the mathematical world: they are presented by ODE coupled with their Liouville equation, but they belong neither to Newtonian nor to quantum physics since they are capable to violate the second law of thermodynamics. That suggests that machines could not simulate intuition-based intelligence if they are composed only of physical parts, but without digital components. Nevertheless it was found such quantum-classical hybrids, [1], that simulates some of self-supervised systems. The main achievement of this work is a demonstration that self-supervised systems can solve NP-complete problems in polynomial time by replacing an enumeration of exponentially large number of possible choices with a short cut provided by a non-Newtonian and non-quantum nature of self-supervised systems.
Category:Combinatorics and Graph Theory

Sequences of Primes Obtained by the Method of Concatenation (Collected Papers)

The purpose of this book is to show that the method of concatenation can be a powerful tool in
number theory and, in particular, in obtaining possible infinite sequences of primes.
Category:Combinatorics and Graph Theory

International Journal of Mathematical Combinatorics, Vol. 1/2016

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original researchpapers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.
Category:Combinatorics and Graph Theory

Binding Number of Some Special Classes of Trees

The binding number of a graph G = (V,E) is deﬁned to be the minimum of |N(X)|/|X| taken over all nonempty set X ⊆ V (G) such that N(X) 6= V (G). In this article, we explore the properties and bounds on binding number of some special classes of trees.
Category:Combinatorics and Graph Theory

In this paper, let (α,α∗) be Bertrand curve pair, when the unit Darboux vector of the α∗ curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated. These values are expressed depending upon the α curve. Besides, we illustrate example of our main results.
Category:Combinatorics and Graph Theory

On Net-Regular Signed Graphs

In this paper, we obtained the characterization of net-regular signed graphs and also established the spectrum for one class of heterogeneous unbalanced net-regular signed complete graphs.
Category:Combinatorics and Graph Theory

Mathematical Combinatorics (International Book Series)

The Mathematical Combinatorics (International Book Series) is a fully refereed international book series with ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.
Category:Combinatorics and Graph Theory

P vs NP Problem Solutions Generalized

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.
Category:Combinatorics and Graph Theory

On Bipolar Single Valued Neutrosophic Graphs

In this article, we combine the concept of bipolar neutrosophic set and graph theory. We introduce the notions of bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, complete bipolar single valued neutrosophic graphs, regular bipolar single valued neutrosophic graphs and investigate some of their related properties
Category:Combinatorics and Graph Theory

Exact Minimum Lower Bound Algorithm for Traveling Salesman Problem

The minimum-travel-cost algorithm is a dynamic programming algorithm to compute an exact and deterministic lower bound for the general case of the traveling salesman problem (TSP). The algorithm is presented with its mathematical proof and asymptotic analysis. It has a (n2) complexity. A program is developed for the implementation of the algorithm and successfully tested among well known TSP problems.
Category:Combinatorics and Graph Theory

Mathematics for Everything with Combinatorics on Nature. a Report on the Promoter Dr. Linfan Mao of Mathematical Combinatorics

The science's function is realizing the natural world, developing our society in coordination with natural laws and mathematics provides the quantitative tool and method for solving problems helping with that understanding. Generally, understanding a natural thing by mathematical ways or means to other sciences are respectively establishing mathematical model on typical characters of it with analysis first, and then forecasting its behaviors, and finally, directing human beings for hold on its essence by that model.
Category:Combinatorics and Graph Theory

Interval Valued Neutrosophic Graphs

The notion of interval valued neutrosophic sets is a generalization of fuzzy sets, intuitionistic fuzzy sets, interval valued fuzzy sets, interval valued intuitionstic fuzzy sets and single valued neutrosophic sets. We apply for the first time the concept of interval valued neutrosophic sets, an instance of neutrosophic sets, to graph theory. We introduce certain types of interval valued neutrosophc graphs (IVNG) and investigate some of their properties with proofs and examples.
Category:Combinatorics and Graph Theory

Single Valued Neutrosophic Graphs

The notion of single valued neutrosophic sets is a generalization of fuzzy sets, intuitionistic fuzzy sets. We apply the concept of single valued neutrosophic sets, an instance of neutrosophic sets, to graphs. We introduce certain types of single valued neutrosophic graphs (SVNG) and investigate some of their properties with proofs and examples.
Category:Combinatorics and Graph Theory

International Journal of Mathematical Combinatorics, Vol. 4, 2015

A Proof of the Four Color Theorem by Induction

Authors:Quang Nguyen VanComments: 14 pages. We have other one written in tex.file

We choose one of four colors as a temporary color for all regions that have not been colored,and made the initial conditions corresponding to The Four Color Theorem. If these conditions hold for any n regions figure, then they will hold for n + 1 regions figure - formed n regions figure by adding next region. By induction, step by step we have proved The Four Color theorem successfully on paper (in 2015).
Category:Combinatorics and Graph Theory

Abstract: In a previous paper we described a method to represent graph information as a single numerical value by distinctly labeling each of its vertices with unique primes. In this paper, we modify the previous approach to again represent a graph as a single numeric value, we log transform this value and approximate it with an optimum value which if minimized by appropriate prime labeling of the graph should allow us to compare it with another graph on which an identical algorithm is implemented. Identical optimum value minima may be expected to indicate graph isomorphism.
Category:Combinatorics and Graph Theory

Isomorphism of Graphs using Ordered Adjacency List

In this paper we develop a novel characterization for isomorphism of graphs. The characterization is obtained in terms of ordered adjacency lists to be associated with two given labeled graphs. We show that the two given labeled graphs are isomorphic if and only if their associated ordered adjacency lists can be made identical by applying the action of suitable transpositions on any one of these lists. We discuss in brief the complexity of the algorithm for deciding isomorphism of graphs and show that it is of the order of the cube of number of the number of edges.
Category:Combinatorics and Graph Theory

A Prime Number Based Strategy to Label Graphs and Represent Its Structure as a Single Numerical Value

We present a simple theoretical strategy to represent using a single numerical
value “A” called the prime vertex labeling Adjacency value, all structural information
encoded in a graph. This strategy has the potential to allow us to reconstruct the graph in
its entirety based on a single number. To do so we assume that we have access to a large
list of prime numbers which are infinite in number. This method will allow us to store
graph backbone as a numerical value for retrieval and re-use and may also allow
development of algorithms that exploit this representation feature as shortcut to address
graph isomorphism.
Category:Combinatorics and Graph Theory

Counting 2-way Monotonic Terrace Forms over Rectangular Landscapes

Authors:Richard J. MatharComments: 31 Pages. Includes a Java program licensed under the LGPL v3.0.

A terrace form assigns an integer altitude to each point of a finite two-dimensional square grid
such that the maximum altitude difference between a point and its four neighbors is one.
It is 2-way monotonic if the sign of this altitude difference is zero or one for steps to the East or steps
to the South. We provide tables for the number of 2-way monotonic terrace forms as a function of grid size
and maximum altitude difference, and point at the equivalence to the number of 3-colorings of the grid.
Category:Combinatorics and Graph Theory

A Class of Multinomial Permutations Avoiding Object Clusters

Authors:Richard J. MatharComments: Pages 9 to 21 are a JAVA program distributed under the LGPL v3.

The multinomial coefficients count the number of ways (of permutations) of
placing a number of partially distinguishable objects on a line, taking ordering
into account. A well-known two-parametric family of counts arises if there are objects
of c distinguishable colors and m objects of each color, m*c objects in total, to be placed on line.
In this work we propose an algorithm to count the permutations where
no two objects of the same color appear side-by-side on the line. This eliminates
all permutations with "clusters" of colors. Essentially we represent filling
the line sequentially with objects as a tree of states where each node
matches one partially filled line. Subtrees are merged if they have the same
branching structure, and weights are assigned to nodes in the tree keeping track
of how many mergers take place. This is implemented in a JAVA program; numerical results
confirm Hardin's earlier counts for this kind of restricted permutations.
Category:Combinatorics and Graph Theory

Authors:Dainis ZepsComments: 16 Pages. The article is a Mathematica notebook

In this article we explore 4-critical graphs using Mathematica. We generate graph patterns according [1]. Using the base graph, minimal planar multiwheel and in the same time minimal according projective pattern built multiwheel, we build minimal multiwheels according [1], We forward two conjectures according graphs augmented according considered patterns and their 4-criticallity, and argue them to be proved here if the paradigmatic examples of this article are accepted to be parts of proofs.
Category:Combinatorics and Graph Theory

Ripà’s Conjectures on the K-Dimensions 3 X 3 X … X 3 Dots Problem

The classic thinking problem, the “Nine Dots Puzzle”, is widely used in courses on creativity and appears in a lot of games magazines. Here are two mutually exclusive conjectures about the generic solution of the problem of the 3k dots spread to 3 X 3 X … X 3 points, in a k-dimensional space.
Category:Combinatorics and Graph Theory

A Computer Program to Solve Water Jug Pouring Puzzles.

We provide a C++ program which searches for the smallest number of pouring steps
that convert a set of jugs with fixed (integer) capacities and some initial known (integer)
water contents
into another state with some other prescribed water contents. Each step
requires to pour one jug into another
without spilling
until either the source jug is empty or the
drain jug is full-because the model assumes the jugs have irregular shape and no
further marks.
The program simply
places the initial jug configuration at the root of the tree
of state diagrams and deploys the branches (avoiding loops) recursively by
generating all possible states from known states in one pouring step.
Category:Combinatorics and Graph Theory

The n X n X n Points Problem Optimal Solution

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° and 45° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach, that significantly reduces the number of straight lines connected at their end-points necessary to join all the n3 dots, for any n > 5. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference h_u(n) − h_l(n) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), if n > 1.
Category:Combinatorics and Graph Theory

An Efficient Method for Computing Ulam Numbers

The Ulam numbers form an increasing sequence beginning 1,2 such that each subsequent number can be uniquely represented as the sum of two smaller Ulam numbers. An algorithm is described and implemented in Java to compute the first billion Ulam numbers.
Category:Combinatorics and Graph Theory

A Note on Erdős-Szekeres Theorem

Erdős-Szekeres Theorem is proven. The proof is very similar to the original given by
Erdős and Szekeres. However, it explicitly uses properties of binary trees to prove and
visualize the existence of a monotonic subsequence. It is hoped that this presentation
is helpful for pedagogical purposes.
Category:Combinatorics and Graph Theory

P vs NP: Solutions of the Traveling Salesman Problem

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique.
Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..
Category:Combinatorics and Graph Theory

The Eigen-3-Cover Ratio of Graphs: Asymptotes, Domination and Areas

The separate study of the two concepts of energy and vertex coverings of graphs has opened many avenues of research. In this paper we combine these two concepts in a ratio, called the eigen-3-cover ratio, to investigate the domination effect of the subgraph induced by a vertex 3-covering of a graph (called the 3-cover graph of ), on the original energy of , where large number of vertices are involved. This is referred to as the eigen-3-cover domination and has relevance, in terms of conservation of energy, when a molecule’s atoms and bonds are mapped onto a graph with vertices and edges, respectively. If this energy-3-cover ratio is a function of , the order of graphs belonging to a class of graph, then we discuss its horizontal asymptotic behavior and attach the graphs average degree to the Riemann integral of this ratio, thus associating eigen-3-cover area with classes of graphs. We found that the eigen-3-cover domination had a strongest effect on the complete graph, while this eigen-3-cover domination had zero effect on star graphs. We show that the eigen-3-cover asymptote of discussed classes of graphs belong to the interval [0,1], and conjecture that the class of complete graphs has the largest eigen-3-cover area of all classes of graphs.
Category:Combinatorics and Graph Theory

Negating Four Color Theorem with Neutrosophy and Quad-stage Method

With the help of Neutrosophy and Quad-stage Method, the proof for negation of “the four color theorem” is given. In which the key issue is to consider the color of the boundary, thus “the two color theorem” and “the five color theorem” are derived to replace "the four color theorem".
Category:Combinatorics and Graph Theory

Replacements of recent Submissions

A New Algebraic Approach to the Graph Isomorphism and Clique Problems

Authors:Roman GalayComments: 12 Pages. The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually this fundamental principle can be efficiently applicable in Computational Mathematics and Complexity Theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) on a class of instances is often too difficult to determine, although we may also assume the existence of a variety of algorithms for NP-complete problems that are both correct and polynomial-time on all the instances from a given class (where the given problem remains NP-complete), but whose correctness or/and polynomial-time complexity on the class is impossible to prove as an example for Gödel's theorems. However, supposedly such algorithms should possess a certain complicatedness of processing the input data and treat it in a certain algebraically “entangled” manner. The same algorithmic analysis in fact concerns all the other significant problems and subclasses of NP, such as the graph isomorphism problem and its associated complexity class GI.
The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.
Category:Combinatorics and Graph Theory

A New Algebraic Approach to the Graph Isomorphism and Clique Problems

Authors:Roman GalayComments: 8 Pages. The following short article offers a couple of algebraically "entangled" polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.

As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually this fundamental principle can be efficiently applicable in Computational Mathematics and Complexity Theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) on a class of instances is often too difficult to determine, although we may also assume the existence of a variety of algorithms for NP-complete problems that are both correct and polynomial-time on all the instances from a given class (where the given problem remains NP-complete), but whose correctness or/and polynomial-time complexity on the class is impossible to prove as an example for Gödel's theorems. However, supposedly such algorithms should possess a certain complicatedness of processing the input data and treat it in a certain algebraically “entangled” manner. The same algorithmic analysis in fact concerns all the other significant problems and subclasses of NP, such as the graph isomorphism problem and its associated complexity class GI.
The following short article offers a couple of algebraically entangled polynomial-time algorithms for the graph isomorphism and clique problems whose correctness is yet to be determined either empirically or through attempting to find proofs.
Category:Combinatorics and Graph Theory

The n X n X n Dots Problem: An Improved “Outside the Box” Upper Bound

In this paper we describe two new patterns, in order to improve the upper bound for the Ripà’s n X n X n points problem, saving a few lines for many values of n. The new upper bound, for any n≥6, becomes h_u(n)=int((3/2*n^2)+int(n/4)-int((n-1)/4)+int((n+1)/4)-int((n+2)/4)+n-2, where int(x)≔floor(x).
Category:Combinatorics and Graph Theory

Envp, Another Prime Number Based Strategy to Encode Graphs

In this paper we show a method to encode graphs with a numerical value that follows unique labeling of each vertex or node and unique labeling of each edge of a graph with unique prime numbers. Each edge is defined as the connectivity between two vertices, therefore two vertices or nodes connected by an edge may be represented by the “ edge-nodes value ” derived by raising the prime number representing the edge to the product of the primes representing the two nodes that are connected by that edge. Multiplying all the “edge-nodes values” of a single graph will represent a unique number albeit very large in majority of cases. Given this unique number called the “Edge-nodes values product”, it is possible to derive the structure of the given graph. This encoding may allow new approaches to graph isomorphism, cryptography, quantum computing, data security, artificial intelligence, etc.
Category:Combinatorics and Graph Theory

On the Division of Plane Figures in Consecutive Prime Parts

In this paper it is discussed the following problem: "A mathematician wants to divide is garden into consecutive prime parts (first in two parts, after in three parts, and so on), only making straight paths, in a simple way (without retracing his own steps), and without going out of his plot of land. In how many parts can the mathematician divide his garden?"
Category:Combinatorics and Graph Theory

The Exact Solution of Gauss’s Problem on the Number of Integer "Points" in a Circular and Spherical "Layers"

In the article, the Gauss’s problem on the number of integer points for a circle and a ball in the framework of an integer lattice is reformulated in an equivalent way and reduces to solving two combinatorial tasks for a circular and spherical "layer" in the framework of Quantum Discrete Space. These tasks are solved using trigonometric functions defined on a set of integers whose range of values is also integers, and other new mathematical tools. It comes not about evaluative solutions, but about exact solutions, which, if necessary, can be transferred to a circle and a ball. In doing so not only specific formulas for determine the exact number of solutions are presented, but also the formulas for enumerating the corresponding pairs and triples of integers. The importance of obtained solutions lies in the fact that they determine the analytical likenesses of not only the circumference and the sphere in the Quantum Discrete Space, but also point to the possibility of constructing of the likenesses of ellipse, cone, hyperboloid and other figures.
Category:Combinatorics and Graph Theory

This paper is concerned with the problem of exact MAP inference in general higher-order
graphical models by means of a traditional linear programming relaxation approach. In fact, the proof that we have developed in this paper is a rather simple algebraic proof being
made straightforward, above all, by the introduction of two novel algebraic tools. Indeed, on the one hand, we introduce the notion of delta-distribution which merely stands for the difference of two arbitrary probability distributions, and which mainly serves to alleviate the sign constraint inherent to a traditional probability distribution. On the other hand, we develop an approximation framework of general discrete functions by means of an orthogonal projection expressing in terms of linear combinations of function margins with respect to a given collection of point subsets, though, we rather exploit the latter approach for the purpose of modeling locally consistent sets of discrete functions from a global perspective. After that, as a first step, we develop from scratch the expectation optimization framework which is nothing else than a reformulation, on stochastic grounds, of the convex-hull approach, as a second step, we develop the traditional LP relaxation of such an expectation optimization approach, and we
show that it enables to solve the MAP inference problem in graphical models under rather general assumptions. Last but not least, we describe an algorithm which allows to compute an exact MAP solution from a perhaps fractional optimal (probability) solution of the proposed LP relaxation.
Category:Combinatorics and Graph Theory

Traveling Salesman Problem Solved with Zero Error

The traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The general approach to solving the different types of NP problems is the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. In the salesman problem, the first step is to arrange the data in the problem in increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The approach in this paper is different from the author's previous approach (viXra:1505.0167) in which the needed distances not among the least ten distances were added to the least ten distances before route construction began. In this paper, one starts with only the least ten distances and only if a needed distance is not among the set of the least ten distances, would one consider distances greater than those in the set of the ten least distances.
The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in workforce project management and hiring, as well as in a country's workforce needs and immigration quota determination. Since an approach that solves one of these problems can also solve other NP problems, and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied
Category:Combinatorics and Graph Theory

New Lower Bounds for Van Der Waerden Numbers Using Genetic Algorithm

Genetic algorithm is a good tool for finding the global minimum in many discrete problems. In particular, it has proven itself in problems where there is no any apriori information about the possibilities of narrowing the search, or the specifics of the problem do not involve such. This work describes the procedure of using a genetic algorithm as applied to the search of van der Waerden numbers. Some new lower bounds of van der Waerden numbers were found using this procedure. Using the genetic algorithm, the author has found the best lower estimates for the Van der Waerden number W (7, 4), W (5, 5), W (6, 5), W (5, 6).
Category:Combinatorics and Graph Theory

Hamiltonian Paths in Graphs

In this paper, we explore the connections between graphs and Turing
machines. A method to construct Turing machines from a general
undirected graph is provided. Determining whether a Hamiltonian
cycle exists is now shown to be equivalent to solving the halting
problem.
We investigate applications of the halting problem to problems in number theory.
A modified version of the classical Turing machine is now developed to
solve certain classes of computational problems.
Category:Combinatorics and Graph Theory

P vs NP Problem Solutions Generalized

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly and completely, the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.
Category:Combinatorics and Graph Theory

P vs NP Problem Solutions Generalized

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly and completely, the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.
Category:Combinatorics and Graph Theory

P vs NP Problem Solutions Generalized

This paper covers the principles and procedures for producing the solution of a problem given the procedure for checking the solution of the problem and vice versa. If a problem can be checked in polynomial time, it can be solved in polynomial time, provided a complete checking procedure is available. From a point A, if one uses one's feet to measure a certain distance by counting steps forwards to a point B, and one wants to check the correctness of the measurement, one would count backwards from the point B using one's feet to see if one returns to exactly the point A. If one returns to A, the forward counting is correct, otherwise it is incorrect. If one counted backwards first from the point B to the point A, one could also count forwards from A to B. Before computers were used in filing taxes in the United States, when one prepared a tax return and wanted to check for arithmetic errors, one would reverse the arithmetic steps from the last arithmetic statement backwards all the way to the first entry on the tax form; and if one obtains a zero after reversing the steps, one was sure that there were no arithmetic errors on the tax form (That is, one began with zero entry going forward and one returned with a zero entry). So also, if one is able to check quickly the correctness of the solution to a problem, one should also be able to produce the solution of the problem by reversing the steps of the checking process while using opposite operations in each step. If a complete checking process is available, the solution process can be obtained by reversing the steps of the checking while using opposite operations in each step. In checking the correctness of the solution to a problem, one should produce the complete checking process which includes the end of the problem and the beginning of the problem. Checking only the final answer or statement is incomplete checking. Since the solution process and the checking process are inverses of each other, knowing one of them, one can obtain the other by reversing the steps while using opposite operations. To facilitate complete checking, the question should always be posed such that one is compelled to show a checking procedure from which the solution procedure can be deduced. Therefore P is always equal to NP.
Category:Combinatorics and Graph Theory

In a previous paper we described a method to represent graph information as a single numerical value by distinctly labeling each of its vertices with unique primes. In this paper, we modify the previous approach to again represent a graph as a single numeric value, we log transform this value and approximate it with an optimum value which if minimized by appropriate prime labeling of the graph should allow us to compare it with another graph on which an identical algorithm is implemented. Identical optimum value minima is a necessary but not sufficient condition for graph isomorphism.
Category:Combinatorics and Graph Theory

In a previous paper we described a method to represent graph information as a single numerical value by distinctly labeling each of its vertices with unique primes. In this paper, we modify the previous approach to again represent a graph as a single numeric value, we log transform this value and approximate it with an optimum value which if minimized by appropriate prime labeling of the graph should allow us to compare it with another graph on which an identical algorithm is implemented. Identical optimum value minima may be expected to indicate graph isomorphism.
Category:Combinatorics and Graph Theory

Isomorphism of Graphs using Ordered Adjacency List

In this paper we develop a novel characterization for isomorphism of graphs. The characterization is obtained in terms of ordered adjacency lists to be associated with two given labeled graphs. We show that the two given labeled graphs are isomorphic if and only if their associated ordered adjacency lists can be made identical by applying the action of suitable transpositions on any one of these lists. We discuss in brief the complexity of the algorithm for deciding isomorphism of graphs and show that it is of the order of the cube of number of the number of edges.
Category:Combinatorics and Graph Theory

A Prime Number Based Strategy to Label Graphs and Represent Its Structure as a Single Numerical Value

We present a simple theoretical strategy to represent using a single numerical value “A” called the prime vertex labeling Adjacency value product, all structural information encoded in a graph. This strategy has the potential to allow us to reconstruct the graph in its entirety based on a single number. To do so we assume that we have access to a large list of prime numbers which are infinite in number. This method will allow us to store graph backbone as a numerical value for retrieval and re-use and may also allow development of algorithms that exploit this representation feature as shortcut to address graph isomorphism.
Category:Combinatorics and Graph Theory

A Prime Number Based Strategy to Label Graphs and Represent Its Structure as a Single Numerical Value

We present a simple theoretical strategy to represent using a single numerical value “A” called the prime vertex labeling Adjacency value product, all structural information encoded in a graph. This strategy has the potential to allow us to reconstruct the graph in its entirety based on a single number. To do so we assume that we have access to a large list of prime numbers which are infinite in number. This method will allow us to store graph backbone as a numerical value for retrieval and re-use and may also allow development of algorithms that exploit this representation feature as shortcut to address graph isomorphism.
Category:Combinatorics and Graph Theory

Counting 2-way Monotonic Terrace Forms over Rectangular Landscapes

A terrace form assigns an integer altitude to each point of a finite two-dimensional square grid
such that the maximum altitude difference between a point and its four neighbors is one.
It is 2-way monotonic if the sign of this altitude difference is zero or one for steps to the East or steps
to the South. We provide tables for the number of 2-way monotonic terrace forms as a function of grid size
and maximum altitude difference, and point at the equivalence to the number of 3-colorings of the grid.
Category:Combinatorics and Graph Theory

Ripà’s Conjectures on the K-Dimensions 3 X 3 X … X 3 Dots Problem

The classic thinking problem, the “Nine Dots Puzzle”, is widely used in courses on creativity and appears in a lot of games magazines. Here are two mutually exclusive conjectures about the generic solution of the problem of the 3^k dots spread to 3 X 3 X … X 3 points, in a k-dimensional space.
Category:Combinatorics and Graph Theory

Ripà’s Conjectures on the K-Dimensions 3 X 3 X … X 3 Dots Problem

The classic thinking problem, the “Nine Dots Puzzle”, is widely used in courses on creativity and appears in a lot of games magazines. Here are two mutually exclusive conjectures about the generic solution of the problem of the 3^k dots spread to 3 X 3 X … X 3 points, in a k-dimensional space.
Category:Combinatorics and Graph Theory

A Computer Program to Solve Water Jug Pouring Puzzles.

Authors:Richard J. MatharComments: 33 Pages. Added more references and more explicit solutions in Version 2.

We provide a C++ program which searches for the smallest number of pouring steps
that convert a set of jugs with fixed (integer) capacities and some initial known (integer)
water contents
into another state with some other prescribed water contents. Each step
requires to pour one jug into another
without spilling
until either the source jug is empty or the
drain jug is full-because the model assumes the jugs have irregular shape and no
further marks.
The program simply
places the initial jug configuration at the root of the tree
of state diagrams and deploys the branches (avoiding loops) recursively by
generating all possible states from known states in one pouring step.
Category:Combinatorics and Graph Theory

The n X n X n Points Problem Optimal Solution

We provide an optimal strategy to solve the n X n X n points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach, that significantly reduces the number of straight lines connected at their end-points necessary to join all the n3 dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference hu(n) − hl(n) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ n ∙ (n + 3), for any n > 1.
Category:Combinatorics and Graph Theory

An Efficient Method for Computing Ulam Numbers

The Ulam numbers form an increasing sequence beginning 1,2 such that each subsequent number can be uniquely represented as the sum of two smaller Ulam numbers. An algorithm is described and implemented in Java to compute the first billion Ulam numbers.
Category:Combinatorics and Graph Theory

P vs NP: Solutions of the Traveling Salesman Problem

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique. Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..
Category:Combinatorics and Graph Theory

P vs NP: Solutions of the Traveling Salesman Problem

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved may not be unique. Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..
Category:Combinatorics and Graph Theory

P vs NP: Solutions of the Traveling Salesman Problem

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique.
Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..
Category:Combinatorics and Graph Theory

P vs NP: Solutions of the Traveling Salesman Problem

For one more time, yes, P is equal to NP. For the first time in history, the traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, 100 cities, or 1000 cities, and return to the home base city. The formerly NP-hard problem is now NP-easy problem.
The general approach to solving the different types of NP problems are the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem. The first step is to arrange the data in the problem in increasing or decreasing order. In the salesman problem, the order will be increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique.
Since an approach that solves one of these problems can also solve other NP problems. and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. If all NP problems can be solved, then all NP problems are P problems and therefore, P is equal to NP. The CMI Millennium Prize requirements have been satisfied..
Category:Combinatorics and Graph Theory